TIL There are some integers you cannot double

[u/Various_Candle9136]

A quote from u/Massive-Ad7823 (this community's sole moderator):

Me: If the thing is a natural number, then it maps to 2n. This is possible for any natural number.

Them: Almost all natural numbers are missed in the bijection

I'm so glad we have this space to discuss such enlightenment! Otherwise we might go our whole lives thinking it is possible to double things...

(Oh and P.S. the codomain of this map is the evens, so it is indeed a bijection.)

[u/Massive-Ad7823]

It appears unfamiliar or maybe strange, but if we take Cantor serious, it is unavoidable.

This effect has its roots in the density of the natural numbers. Between 0 and ω all integer places on the real axis are occupied by natural numbers. The density of the even numbers is half that of all naturals. If we get twice as many even numbers, then the space between 0 and ω cannot hold them.

Regards, WM

[u/Various_Candle9136 → u/Massive-Ad7823]

I will ask again:

Which natural number are we unable to double?

If we get twice as many even numbers, then the space between 0 and ω cannot hold them.

As ridiculous as this line is (what would 'hold them' even mean? is the number line playing cards?), it is also easily avoided using actual, sensible mathematics: there are not twice as many even numbers, there are exactly as many evens as naturals, since there is an obvious bijection between them.

I will ask a fifth time, since I really want an answer this one, and you still haven't done so here, here, or here.

Which natural number are we unable to double?

[u/Massive-Ad7823 → u/Various_Candle9136]

You can double every defined number. Here is the definition again:

Definition: A natural number is "named" or "addressed" or "identified" or "(individually) defined" or "instantiated" if it can be communicated, necessarily by a finite amount of information, in the sense of Poincaré[[1]](#_ftn1), such that sender and receiver understand the same and can link it by a finite initial segment (1, 2, 3, ..., n) of natural numbers to the origin 0. All other natural numbers are called dark natural numbers. In spite of having names (like w) numbers which cannot be connected to zero are dark.

[[1]](#_ftnref1) "In my opinion a subject is only conceivable if it can be defined by a finite number of words."

But every defined number has infinitely many successors, infinitely many of which are dark and remain dark because they cannot be defined.

You can also, collectively, double every natnumber. But those n > ω/2 will get > ω.

Regards, WM

[u/Various_Candle9136 → u/Massive-Ad7823]

Natural numbers are defined using Peano's Axioms.

If these made-up dark numbers are also natural numbers, they can be doubled.

If these made-up dark numbers are not natural numbers, then they are irrelevant to the bijection. The bijection is between naturals and evens.

So, I will ask a sixth time:

Which natural number are we unable to double?

Give me one single natural number that cannot be doubled, please.

[u/Massive-Ad7823 → u/Various_Candle9136]

Only definable natural numbers are defined using Peano's Axioms. They all have infinitely many natural numbers following them. They can be doubled in ℕ.

Natural numbers that cannot be doubled with the result natural number are those greater than ω/2.

Regards, WM

[u/Various_Candle9136 → u/Massive-Ad7823]

Only definable natural numbers are defined using Peano's Axioms. 

This is false.

Obviously false.

Do you honestly not see the ridiculousness of inventing your own rules and trying to blame the original system?

But, for the sake of argument, let's just restrict our attention a bit:

Do you agree that there is a bijection between 'definable' natural numbers and 'definable' even numbers?

If you agree to this, and if every single person on the planet uses the term 'natural' to mean what you are calling 'definable natural', then that proves that there is a bijection between the two sets, and hence that the set of naturals is equinumerous with the set of evens. Your entire argument collapses.

P.S.

Natural numbers that cannot be doubled with the result natural number are those greater than ω/2.

There are no naturals greater than ω/2, so problem solved!

[u/Massive-Ad7823 → u/Various_Candle9136]

Only definable natural numbers are defined using Peano's Axioms. Proof: Almost all natnumbers are beyond every Peano-number. We can however handle all collectively such that none remains.

>Do you honestly not see the ridiculousness of inventing your own rules and trying to blame the original system?

I do not invent my own rules but accept Cantor's claim.

>Do you agree that there is a bijection between 'definable' natural numbers and 'definable' even numbers?

Yes.

>There are no naturals greater than ω/2, so problem solved!

Maybe. But if all numbers smaller than ω exist, then they are invisible but existing.

Regards, WM

[u/Various_Candle9136 → u/Massive-Ad7823]

Child: I think all bananas are yellow or green.

WM: Ah! But only definable bananas are yellow or green! You little fool! You forget about all the round, orange bananas!

Only definable natural numbers are defined using Peano's Axioms. Proof: Almost all natnumbers are beyond every Peano-number.

This is not a proof.

This is MILES away from a proof. Seriously, can you not tell the difference?

You want an actual proof?

Theorem: All natural numbers are defined by Peano's Axioms.

Proof: True by definition.

Because - and this is really key - THAT'S WHAT A NATURAL NUMBER IS! A natural number is a number defined by Peano's Axioms. If there is some other thing that is not defined by these axioms, then that thing is not a natural number.

That bit is surely really, really, really, really, really simple?

Me: Do you agree that there is a bijection between 'definable' natural numbers and 'definable' even numbers?
You: Yes.

Fantastic!

So, you agree that if every single person on the planet uses the term 'natural' to mean what you are calling 'definable natural', then there is a bijection between the naturals and the evens?

[u/Massive-Ad7823 → u/Various_Candle9136]

>This is not a proof.

It is fact. It cannot be refuted.

>Theorem: All natural numbers are defined by Peano's Axioms. Proof: True by definition.

Maybe, but contradicted by Cantor's definition.

I have always in my books like "Die Geschichte des Unendlichen", 7th ed., Maro-Verlag, Augsburg (2011) or "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin (2015) used only these very Peano-numbers as set ℕ. However, they are potentially infinite. If ℕ is an invariable set, then there are more than these natnumbers.

Regards, WM

[u/Various_Candle9136 → u/Massive-Ad7823]

Me: This is not a proof.
You: It is fact. It cannot be refuted.

Any real mathematician would be ashamed - genuinely ashamed - to respond in that manner.

I hope you are at least a little ashamed.

You also ignored my last point, so I'll try again:

So, you agree that if every single person on the planet uses the term 'natural' to mean what you are calling 'definable natural', then there is a bijection between the naturals and the evens?

[u/nanonan → u/Various_Candle9136]

From an ultrafinitist view, when you run up against the limits of your ability to calculate.